Micro-Modeling of Polymer–Masonry Wall Composites Under In-Plane Loading

Abstract

Fiber-reinforced polymers (FRPs) are effective for strengthening masonry walls. Debonding at the polymer–masonry interface is a major concern, requiring further investigation into interface behavior. This study utilizes detailed micro-modeling finite element (FE) analysis to predict failure mechanisms and analyze the behavior of brick masonry walls strengthened with externally bonded carbon fiber-reinforced polymer (CFRP) under in-plane loading. The research investigates three CFRP strengthening configurations (X, I, and H). The FE model incorporates the nonlinear behavior of brick masonry components using the Concrete Damage Plasticity (CDP) model and uses a cohesive interface approach to model unit–mortar interfaces and the bond joints between masonry and CFRPs. The results demonstrate that diagonal CFRP reinforcement enhances the ductility and capacity of masonry wall systems. The FE model accurately captures the crack propagation, fracture mechanisms, and shear strength of both unreinforced and reinforced walls. The study confirms that the model can reliably predict the structural behavior of these composite systems. Furthermore, the study compares predicted shear strengths with established design equations, highlighting the ACI 440.7R-10 and CNR-DT 200/2013 models as providing the most accurate predictions when compared to experimental results.

1. Introduction

Several numerical and experimental studies have been reported in the recent literature that investigate the efficiency of the application of carbon fiber-reinforced polymers (CFRPs) on the seismic response of masonry structures subjected to in-plane or out-of-plane loading [1,2,3,4]. For the strengthening of masonry buildings, many experimental studies used CFRP sheets/fabric as an external strengthening system for retrofitting masonry walls [5,6,7]. All the research reported in the literature confirmed that externally bonded FRP composites can enhance the shear strength and deformation capacity of masonry walls subjected to both out-of-plane and in-plane loading [8,9,10]. Tests of masonry walls/panels externally bonded with FRP strips and subjected to diagonal monotonic compression to model shear action also showed large improvements in the strength of the masonry walls/panels [11].

Several different FRP strengthening schemes have been tested to address the shear failure mechanism [12]. The improvement of the overall behavior of structural masonry under shear can be achieved by the addition of horizontal and vertical reinforcement to the bed joint [13]. The monotonic shear loads of masonry walls with FRPs bonded parallel to the bed joints, in diagonal directions, or across the full wall have shown large increases in lateral resistance [14]. The orientation of FRPs perpendicular to the bed joints leads to an increase in bond strength compared to an orientation parallel to the bed joints [15]. The effect of vertical and horizontal reinforcement distribution on the shear strength of concrete block masonry was investigated in [16] based on diagonal compression tests. They concluded that the better configuration for reinforcement arrangement was the combination of vertical and horizontal reinforcements, leading simultaneously to the improvement of shear strength and ductility. Therefore, to further explore the influence of reinforcement arrangement on the shear resistance of masonry walls, this study investigates three distinct CFRP configurations.

Masonry walls are commonly modeled using either heterogeneous or homogeneous approaches [17,18]. Homogeneous models treat the masonry as an equivalent continuum, suitable for large-scale analysis where local failure modes are not critical. However, these models cannot capture the influence of mortar joints, a primary source of weakness and nonlinearity. In the second approach (micro-modeling), the bricks and mortar are modeled separately through their respective constitutive laws, and the interaction between them is considered. This modeling approach represents masonry units and mortar with their actual thicknesses, treating the unit–mortar interfaces as having negligible thickness. The units and mortar are modeled as continuum elements and unit–mortar interfaces are modeled as discontinuous elements [19,20]. Micro-modeling approaches can be categorized as either detailed (DMM) or simplified (SMM). The main purpose of micro-modeling is to describe the behavior of masonry from the knowledge of component properties and the interface. The micro-modeling strategy is especially suited for structural details where the interaction between the units and mortar is a principal interest, because the mortar joint interfaces are the weakest element in unreinforced masonry structures. While DMM offers the most accurate representation of masonry behavior, its computational demands often limit its application to smaller structural elements [21]. The literature has emphasized the importance of introducing all the failure mechanisms of masonry in the simulation in order to explain its behavior in terms of ultimate load and ductility, whereas the damage is usually concentrated in the mortar–joint interface [22,23].

Different modes of failure, including crushing failure, diagonal shear failure, and CFRP reinforcements debonding from the masonry surface occur in strengthened reinforced walls. In FRP-strengthened masonry walls under in-plane shear loading, debonding is the most common failure and one of the undesirable failure mechanisms. If this rupture occurs, the total tensile capacity of the reinforcement cannot be exploited. Generally, this failure occurs mostly when the bonding between the reinforcement composite and masonry interface is lost. Similarly, the debonding of CFRP composites from FRP-strengthened masonry walls is associated either with the CFRP cut-off location (discontinuity of CFRP composites) or with crack location (discontinuity of mortar). Both lead to normal and/or shear stress concentration at the CFRP-to-masonry interface.

Although much research has focused on FRP-to-concrete bonding [24], recent studies [25,26,27,28] highlight the importance of understanding FRP-to-masonry bond behavior, as debonding is a primary failure mode that limits the effectiveness of CFRP strengthening. Also, these experiments proved that debonding mechanisms are key to understanding failure modes in FRP-strengthened masonry structures. Therefore, it is critical to study this type of failure to fully utilize the high-strength and lightweight benefits of CFRPs as a strengthening material for building and construction.

While finite element modeling has become a powerful tool for analyzing masonry structures, the existing models often have limitations in terms of accurately predicting failure behavior. Many models rely on simplified material models that do not fully capture the nonlinear behavior of masonry under different loading conditions [29]. Furthermore, accurately modeling the interfaces between the brick units and mortar joints remains a challenge since these models often assume a perfect bond [30,31,32], neglecting the potential for sliding, separation, and debonding that can significantly influence the overall structural response.

Numerically, a different strategy can be followed to model the diagonal compression test, but this approach assumes a perfect bond between the FRP strip and the masonry [31,33]. Three modeling approaches can be identified from the literature to model bond behavior between FRPs and masonry walls: the direct model approach, interface model approach, and crack band approach [34]. The accurate modeling of FRP–masonry interfaces is crucial for predicting the behavior of FRP-strengthened structures. The failure mechanics, including the crucial FRP–masonry bonds, are not yet fully understood. However, very limited studies have explored the use of micro-modeling techniques to investigate reinforced masonry walls by including the debonding failure mode of CFRP composites.

This study addresses these limitations by developing a detailed micro-mechanical FE model that incorporates nonlinear material behavior and an efficient explicit solver. Interface behavior is modeled using the interface model approach, with negligible thicknesses for the brick–mortar and FRP–masonry interfaces. By carefully calibrating and validating the model against experimental data, we aim to provide a more accurate and reliable tool for predicting the behavior of CFRP-strengthened masonry walls under in-plane loading.

Diagonal compression tests were conducted on unreinforced and CFRP-reinforced masonry wallettes. The experimental results were used to validate a detailed micro-mechanical FE model capable of capturing key failure modes, including debonding. The model’s accuracy is demonstrated through comparison with the experimental results and existing literature, providing a tool for simulating the behavior of unreinforced masonry (URM) and CFRP-reinforced masonry walls.

2. Materials and Methods

The masonry assemblages employed in this experimental program were constructed using perforated bricks and a single type of cement–lime mortar. A mortar mix proportion of 1:1:5 by volume (cement, lime, and sand) was used to build the masonry wallettes. To evaluate the average compressive strength and elastic modulus of the masonry units utilized in the construction of the prisms, three test series of perforated bricks with a density of 1800 kg/m³ and nominal dimensions of (220 × 105 × 55) mm³ were tested in accordance with EN 771-1 [35]. Additionally, the mechanical properties of the mortar were determined at 7 and 28 days after demolding. The flexural and compressive strengths of the mortar specimens, in the shape of 40 × 40 × 160 mm³ prisms, were tested in accordance with EN1015-11 [36] using a universal testing machine, with the average response values reported. The average compressive strengths and elastic moduli for the bricks and mortars were experimentally measured and tabulated in Table 1.

The masonry composite’s compressive and shear mechanical properties (Table 1) were experimentally determined using specimens made with the same constituent materials [21]. Compressive strength (fc) was evaluated on running bond walls according to EN 1052-1 [37]. Compressive fracture energy (G_fc) was estimated from stack bond prisms, calculating the area under the stress–displacement curves [38]. The cohesion (c) and friction angle (Φ) of the bed joints were found using triplet specimens tested per EN 1052-3 [39], applying three load levels to three specimens for each level [21].

Table 1. Mechanical property testing of materials.

Material Types	Dimensions (mm)	Type of Test	Standard Used	Compressive Strength (N/mm2)	Elastic Modulus	Poisson’s Ratio
Mortar (joints)	10	Flexural and compressive strength	EN 1015-11 [36]	3.66	1822	0.18
Perforated bricks	220 × 105 × 55	Compressive strength	EN 771-1 [35]	24.2	10,000	0.2
		Flexural tensile strength
Masonry prism (3 bricks linked by mortar)	220 × 105 × 55	Compressive test	EN 1052-1 [37]	10.83	------	--------
Masonry prism triplet	55 × 105 × 220	Bond strength between mortar and masonry	EN 1052-3 [39]	17.43	-------	--------

Table 1. Mechanical property testing of materials.

Material Types	Dimensions (mm)	Type of Test	Standard Used	Compressive Strength (N/mm2)	Elastic Modulus	Poisson’s Ratio
Mortar (joints)	10	Flexural and compressive strength	EN 1015-11 [36]	3.66	1822	0.18
Perforated bricks	220 × 105 × 55	Compressive strength	EN 771-1 [35]	24.2	10,000	0.2
		Flexural tensile strength
Masonry prism (3 bricks linked by mortar)	220 × 105 × 55	Compressive test	EN 1052-1 [37]	10.83	------	--------
Masonry prism triplet	55 × 105 × 220	Bond strength between mortar and masonry	EN 1052-3 [39]	17.43	-------	--------

For this study, a unidirectional carbon fiber sheet with a width of 50 mm and a thickness of 0.129 mm, along with an epoxy resin strengthening system (Sika Wrap carbon fiber fabric), were used to reinforce the brick masonry wallettes. The CFRP sheets were adhered using a two-part epoxy adhesive. The mechanical and physical properties of the CFRP reinforcing system and resin were obtained from the manufacturer’s datasheet, as provided in

Table 2. Mechanical and physical properties of CFRP composite.

Young’s Modulus E1 (MPa)	Young’s Modulus E2 (MPa)	Poisson’s Ratio $\mathit{\nu }$12	Shear Modulus G12 (MPa)	Shear Modulus G13, G23 (MPa)	Fiber Density (kg/m3)	Fiber Tensile Strength (MPa)	Thickness of CFRP (mm)	Length/Roll (m)	Rupture Strain %
230,000	230,000	0.3	7000	3000	1820	4000	0.129	≥50	1.70

.

2.1. Tests on Masonry Wallettes (Diagonal Compression Tests)

Masonry wallettes were constructed and tested in accordance with the instructions provided in the RILEM technical recommendations [40]. The test involved applying compressive stress along the diagonals of square masonry specimens measuring 400 × 400 × 105 mm³. Prior to testing, all the wallettes were cured for 28 days. The experimental program of diagonal compression testing included two unreinforced control wallettes (noted as CW) and twelve CFRP-reinforced wallettes with varying configurations of CFRP composites. The goal was to evaluate the reinforcement efficiency and the mode of failure for each reinforcement scheme. Three reinforced wallettes (RWH scheme H) had diagonal strips of carbon fabric bonded orthogonally to the compressive diagonal plus two horizontal cut-off strips parallel to the tensile diagonal. Three reinforced wallettes (RWX scheme X) had two strips bonded to each diagonal plus four cut-off strips bonded orthogonally to the diagonal strips. Three reinforced wallettes (RWI scheme I) had one diagonal strip bonded on the compressed diagonal plus two horizontal cut-off strips bonded orthogonally to the compressive diagonal on each side (see

Table 3. Description of different configurations.

Designation *	Dimensions of Wall (mm)	Ratio (%)	Configuration
URM-CW	400 × 400 × 105	-	-
RWI (scheme I)	400 × 400 × 105	41	One vertical and two horizontals
RWH (scheme H)	400 × 400 × 105	53	One vertical and three horizontals
RWX (scheme X)	400 × 400 × 105	74	Diagonal (X)

* URM: unreinforced masonry; W: wall; R: carbon fiber-reinforced polymer.

). Each wallette was reinforced with a single layer of CFRP on both sides.

CFRP reinforcement was applied seven days after wallette construction. Before applying this composite sheet, the face sides of the masonry wallettes were sanded to eliminate any extra hardened mortar from the joints and loose particles on the surface.

A thick coat of two-component epoxy resin was applied to the masonry surfaces with a paint roller. The reinforced panels were strengthened by Sika Wrap CFRP of 50 mm width. The arrangement of the reinforcing composites has a very important effect on the local behavior of the structure due to the stress distribution and the deformation of the structure. Thus, three different configurations of the retrofit system were investigated to evaluate the reinforcement efficiency and the mode of rupture in the case of each reinforcement. The strengthening schemes and procedure techniques are shown in Figure 1.

The wallettes were placed and centered diagonally between the two plates of a press with the help of two metal shoes that were fixed on two opposite corners; these allowed the transmission of the load to the wall in the vertical direction (see Figure 1). The measurement of displacement was carried out using two displacement comparators which were installed on the diagonals of the wallettes. All force and displacement data were recorded automatically by a data acquisition system and provided by the system. The contact surfaces were regularized by applying a thin layer of mortar between the specimens and the supports.

The shear stress τdt could be calculated according to the state of stress at the center of the wall, following the ASTM 519-02 [41] standard, and with the wall considered as isotropic (see Equation (1)).

$$𝜏dt={\displaystyle \frac{{\mathrm{P}}_{\mathrm{m}\mathrm{a}\mathrm{x}}}{\surd 2\times \mathrm{A}}}$$

(1)

Figure 1. Specimen preparation and CFRP composite installation for unreinforced and reinforced masonry wallettes. (a) Experimental test; (b) numerical model.

Figure 1. Specimen preparation and CFRP composite installation for unreinforced and reinforced masonry wallettes. (a) Experimental test; (b) numerical model.

2.2. Characterizing the Mortar–Brick Interface Behavior

The initial shear strengths of the mortar–brick interfaces, denoted as fv0, were estimated by conducting triplet tests in accordance with the standard UNI EN 1052-3 (2007) [39].

3. Numerical Modeling: Methods and Descriptions

The results from the experimental tests on masonry assemblages were compared to those obtained from the developed numerical model to demonstrate the effectiveness of the proposed approach.

A detailed micro-model (3D) was employed here, considering brick units, mortar, and interfaces. Brick units and mortar joints were modeled as solid blocks. Additionally, the nonlinear behavior of brick and mortar was simulated with a Concrete Damage Plasticity (CDP) model. The unit–mortar interface was modeled as a cohesive interface with zero thickness using the surface-to-surface contact option available in ABAQUS. In the numerical analyses, both the brick and mortar were modeled using eight-node 3D continuum elements with four glass controls and reduced integration (C3D8R).

The geometry of the wallettes and defined interaction surfaces between the units and mortar are also shown in Figure 1.

The mortar layer was modeled using uniformly sized elements with a thickness of 10 mm, reflecting the actual joint thickness. The CFRP composite fabric/sheet was modeled using four-node 3D shell elements (S4R) with three translational and rotational degrees of freedom at each node. This was modeled in ABAQUS using lamina material properties. To capture the behavior of the bonds between the CFRP and the masonry surface, especially those that debonded, an interface model was chosen to accurately model the masonry wallettes strengthened with CFRP composites. The bond behavior was modeled using a layer of interface elements, usually with zero thickness [34].

To replicate the experimental support conditions, the bottom corner of the wallette in the FE model was fully constrained to prevent rigid body motion. Specifically, all translational degrees of freedom (U1, U2, and U3) were fixed at the base nodes of the loading steel plate, representing a fully fixed support. Rotational constraints (UR1, UR2, and UR3) were also applied. Compression loads were applied through displacement control, with a steel loading plate positioned at the loading point to ensure uniform distribution. A vertical displacement of 3.5 mm in total was applied to the upper section of the model.

3.1. Constitutive Material Properties

3.1.1. Concrete Damage Plasticity

The Concrete Damage Plasticity (CDP) model available in the ABAQUS software (2019) package was utilized to simulate the nonlinear response of the brick masonry units and mortar joints. This CDP model has been previously developed and validated for predicting the behavior of concrete and other quasi-brittle materials [42,43]. The primary failure modes captured by this model are tensile cracking and compressive crushing. The CDP model considers separate constitutive relationships for the uniaxial compressive and tensile damage responses of the material, as illustrated in Figure 2a. The CDP approach assumes that the uniaxial compressive and tensile responses of the material, such as concrete or other brittle media, can be characterized by damaged plasticity. All the required model parameters can be derived from the defined compressive and tensile behavioral characteristics of the constituent materials. The modeling of the masonry components (brick and mortar) in the analysis was established through a two-step process. In the first step, the elastic modulus and Poisson’s ratio were introduced as inputs. Subsequently, the damage plasticity model was adopted in the second step to define the nonlinear portion of the stress–strain response. These material properties were assigned to the model through the plasticity table in ABAQUS. For this purpose, the experimentally obtained stress–strain curves for the brick units and mortar, as shown in Figure 2b,c, were utilized as the input data.

The CDP option was defined according to a previous study [44], and the major parameters are presented in

Table 4. Input parameters for finite element modeling.

	Description		Symbol	Units	Value
Parameters from tests	Young’s modulus	Brick	E	MPa	10,000
		Mortar			1880
	Poisson’s ratio	Brick	γ	-	0.2
		Mortar			0.18
	Cohesion		C0	-	0.44
	Friction angle		φ0	rad	0.67
	Residual friction angle		φres	rad
Calibrated parameters	Compressive fracture energy		${Gf}_c$	N/mm
	Interface normal stiffness		knn	N/mm3	40
	Interface shear stiffness		ktt	N/mm3	16
	Tensile strength		ft	MPa	2
	Mode-I fracture energy	Masonry wallette	$G_f^I$	N/mm	0.018
		CFRP interface			1.79
	Mode-II fracture energy	Masonry wallette	$G_f^{II}$	N/mm	0.12
		CFRP interface			1.79
	Masonry dilatancy angle		ψ		0
	Damage initiation normal	Masonry wallette		(N/mm2)	5
		CFRP interface			3
	Damage initiation shear I	Masonry wallette			0.56
		CFRP interface			1.5
	Damage initiation shear II	Masonry wallette			0.56
		CFRP interface			1.5
	Eccentricity parameter		e	-	0.1
	Bi- and unidirectional compression resistance ratio (${\displaystyle \frac{f_{bo}}{f_{co}}}$)		-	-	1.16
	Stress ratio in meridian in tension		k	-	0.67
	Viscosity parameter		ν	m2/s	0.001

. For more details, this model was presented and explained in previous research [45].

Figure 2. Compression and tension inputs of model: (a) Response of concrete to uniaxial loading in tension and compression (CDP) model; (b) compressive behavior and tensile behavior inputs of brick unit; (c) compressive behavior and tensile behavior inputs of mortar.

Figure 2. Compression and tension inputs of model: (a) Response of concrete to uniaxial loading in tension and compression (CDP) model; (b) compressive behavior and tensile behavior inputs of brick unit; (c) compressive behavior and tensile behavior inputs of mortar.

3.1.2. Interface Properties (Cohesive Zone Model)

The cohesive surface-based behavior was employed to model the interfaces between the masonry units and mortar, utilizing a zero-thickness approach. The traction–separation constitutive model was utilized to define the mechanical behavior of these cohesive elements [43]. The linear elastic behavior within the ABAQUS traction–separation model (Figure 3) is defined by an initial stiffness matrix that relates the normal and shear tractions to the corresponding separations at the interface. Specifically, the normal stiffness component is denoted as knn, while the shear stiffness components in the two orthogonal directions are kss and ktt, respectively. The opening of the interfaces in the normal direction (δn) is governed by the Mode-I fracture mechanism, while the shearing modes in the two orthogonal shear directions (δs and δt) are controlled by the Mode-II and Mode-III fracture mechanisms, respectively.

Figure 3. Typical traction–separation behavior (a,b) and fracture mode of masonry joint interfaces [24].

Figure 3. Typical traction–separation behavior (a,b) and fracture mode of masonry joint interfaces [24].

The equivalent stiffness for the joint interfaces can be expressed as a function of the elastic modulus of the masonry units and mortar, as well as the thickness of the mortar joint, as shown in Equations (2) and (3). This formulation allows the traction–separation model to account for the mechanical properties and geometry of the masonry components when defining the initial elastic responses of the interfaces [17,18].

$$k_{nn}={\displaystyle \frac{E_u{E}_m}{h_m\left(E_u-E_m\right)}}$$

(2)

$$k_{ss}=k_t={\displaystyle \frac{G_u{G}_m}{h_m\left(G_u-G_m\right)}}$$

(3)

where h_m represents the thickness of the mortar joint; E_u and E_m are the Young’s moduli of the masonry unit and mortar, respectively; and G_u and G_m are the corresponding shear moduli.

Damage initiation for the cohesive interfaces was governed by the maximum nominal stress ratio criterion available in ABAQUS. Damage initiation was assumed to occur when the function reaches a value of one, as shown in Equation (4):

$${\left({\displaystyle \frac{{\sigma }_n}{{\sigma }_n^0}}\right)}^2+{\left({\displaystyle \frac{{\tau }_n}{{\tau }_s^0}}\right)}^2+{\left({\displaystyle \frac{{\tau }_t}{{\tau }_t^0}}\right)}^2\left\{\begin{array}{c}<1 no failure\\ =1 failure\end{array}\right.$$

(4)

σ_n is the tensile stress; τ_s and τ_t are the shear stresses of the interface; the superscript o represents the initial value; and the subscripts n, s, and t are the directions of the constraint components [46]. Once this criterion was met, the model followed a damage evolution law to describe the gradual degradation of the interface properties.

To accurately model the behavior after contact, the current model defined Coulomb frictional contact behavior. This frictional contact formulation is particularly important for the normal behavior of the contacts, as it helps to avoid element penetration once the cohesive bond has been compromised.

• The constitutive behavior of the CFRP–masonry interface: In addition to the masonry unit–mortar interfaces, the bonds between the CFRP strengthening strips and the masonry surface were also modeled using the cohesive behavior approach, based on the traction–separation constitutive model. This allowed for the representation of the debonding failure mode that can occur at CFRP–masonry interfaces [47].

  $$k_0={\displaystyle \frac{1}{\frac{t_i}{G_i}+\frac{t_c}{G_c}}}$$

  (5)

The initial stiffness of the CFRP–masonry interface is presented in Equation (5), which is a function of the elastic moduli and thicknesses of the CFRP laminate and the masonry substrate. This formulation ensures that the traction–separation model can accurately capture the mechanical behavior of the interface between the reinforcing CFRP and the masonry elements.

Similarly to the treatment of the masonry unit–mortar interfaces, the CFRP–masonry interfaces were also subject to a damage initiation criterion and a damage evolution law. These governed the progressive degradation of the bond properties as the interface experienced increasing deformation and eventual debonding.

3.2. Model Inputs

The input parameters used to present the masonry wallette models were developed based on the experimental results obtained for the individual brick, mortar, and brick–mortar interface components. The diagonal test model (for both unreinforced and CFRP-reinforced masonry wallettes) was simulated using the ABAQUS/Explicit algorithm. This explicit approach is better suited for problems involving complex contact or material behavior, which can cause convergence difficulties in the ABAQUS/Standard implicit solver.

Contact behavior was modeled using "hard" contact to prevent interpenetration.

A correction factor was applied to the Mohr–Coulomb parameters (c and μ) when estimating the lateral in-plane shear strengths of the wallettes in order to account for the difference between local-level and wall-level behavior.

The CDP parameters were defined based on experimental results from the brick and mortar samples. These parameters, shown in Table 4, could be derived from the defined compressive and tensile behavioral characteristics of the constituent materials. The dilation angle (ψ), which influences the material’s plastic flow, was set to 0 degrees, considering the non-associated flow rule of masonry structures and following recommendations in previous research [44].

The initial stiffness of the CFRP–masonry interface was calculated using Equation (5), considering the elastic moduli and thicknesses of the CFRP laminate and the masonry substrate. The fracture energy values for the CFRP interface, listed in Table 4, were carefully selected based on experimental data and the recommendations provided by the CFRP datasheet and other CFRP/concrete bonding research in the literature. For Mode-I fracture energy, the data available in the literature [42] recommend values from 0.005 to 0.2 N/mm for a range of tensile strengths from 0.3 to 0.9 N/mm². For modeling the FRP–masonry interface, we used a value of 1.79 N/mm² [46].

For the tensile fracture energy (Mode-I) of the interface, the data available in the literature [48] recommend values from 0.005 to 0.2 N/mm for a range of tensile strengths from 0.3 to 0.9 N/mm². This was confirmed by Joaquim et al. [49] when they found that, for different types of brick-and-mortar interfaces, the average Mode-I fracture energy was around 0.008 N/mm when the average bond tensile strength was in the order of 2 N/mm². The shear tractions were known as the initial shear strengths, which are related to the cohesion parameter (c) at zero confining stress. It is well recognized that this value depends on the applied vertical load and the mechanical properties of masonry assemblage. The values for cohesion, friction angle, and Mode-I fracture energy in the cohesive interface model were parameterized based on the experimental data derived from the triplet shear tests detailed in our prior work [21]. These tests, conducted on comparable brick–mortar interfaces under varying levels of pre-compression, provided key insights into the materials’ behavior at the joints and directly informed the selection of a Mode-I fracture energy value of 0.018 N/mm.

In this study, the values obtained for the shear tractions ranged from 0.4 up to 0.55 MPa. Additionally, the value of shear fracture energy Mode-II was taken to equal 0.2 for the modeling. Table 4 presents the material properties used in this study for modeling the masonry assemblages, including the brick, mortar, and brick–mortar interface characteristics.

4. Validation and Comparison of Numerical Simulation and Experimental Results

The numerical and experimental shear stress vs. shear strain, in addition to the location of crack initiation and crack propagation in the brick-and-mortar layers, are discussed in the following sections.

4.1. CFRP-Reinforced Masonry Wallette Subjected to Diagonal Compression

Failure Modes

The developed model effectively captures the behavior of CFRP-reinforced masonry wallettes under diagonal compression. It accurately predicts the relationship between shear stress and shear strain, the collapse mechanism, and the failure mode.

Crack patterns observed in both numerical and experimental tests of an unreinforced masonry control wallette (CW) are compared in Figure 4. The control wallette initially cracked along the bed and head mortar joints in a diagonal pattern, followed by a rapid decrease in load capacity. With increasing imposed displacement, the diagonal cracks widened, culminating in the failure of the wallette. This diagonal cracking is consistent with the development of principal tensile stresses perpendicular to the compressed diagonal of the wallette [6].

Figure 5 shows three different types of CFRP-strengthened masonry wallettes under diagonal compression, each with a color-coded stress distribution. The left wallette (scheme I), reinforced with two horizontal CFRP strips, exhibits a high concentration of stress along the center vertical line, with a “pinching” effect around the CFRP strips. This indicates that the CFRP strips effectively transfer the load from the masonry, reducing the overall stress concentration. The center wallette (scheme H), featuring three horizontal CFRP strips, displays a similar pattern but with a slightly reduced stress concentration along the center vertical line, indicating that increasing the amount of CFRP reinforcement further improves load transfer and stress distribution. The right wallette (scheme X), with a more complex reinforcement scheme, using four CFRP strips arranged in a “cross” pattern, demonstrates a more even distribution of stress across the wallette, significantly reducing high-stress concentration areas compared to the first two examples. This suggests that a cross-pattern reinforcement scheme effectively distributes the load and minimizes stress concentration, potentially leading to improved structural performance. Overall, the images visually demonstrate the effectiveness of CFRP reinforcement in improving the load-bearing capacity of masonry wallettes under diagonal compression. By analyzing the stress distribution patterns, engineers can make informed decisions about the optimal design and placement of CFRP reinforcements to enhance the structural integrity of masonry structures.

A comparison of the crack patterns developed in the numerical model and obtained in the experimental tests for the reinforced masonry wallettes is shown in Figure 6.

The RWI wallettes (scheme I) exhibited two types of cracks: tensile and shear diagonal cracks. Diagonal cracking was observed in the mortar joints along the compressed diagonal and near the edges of the CFRP strips. As the lateral load increased, web shell splitting cracks in the brick began at the middle web and spread to the top and bottom of the specimen. Debonding initially occurred at the end of the CFRPs located under the loading points and then extended to the main diagonal crack. As the cracks propagated, a segment of the CFRP reinforcements adjacent to the crack failed, with the extent of this failure increasing with the widening of the crack opening (Figure 6a).

The RWH wallettes (scheme H) did not exhibit diagonal cracking in their mortar joints (Figure 6b). The reinforcement delayed the propagation of diagonal shear cracks and enabled the distribution of stresses across the entire wallette after the onset of major diagonal cracks near the top and bottom of the CFRP reinforcement. Failure was initially governed by vertical tensile cracks in the brick, beginning at the middle web and spreading to the top and bottom of the specimen. Splitting then occurred in the units located under the loading shoes, resulting in the partial debonding of the CFRP in this region . High compressive stress at the top and bottom ends of the wallette, due to the in-plane bending of the walls, damaged the bricks at those locations. Web shell cracks were also observed due to the doubly reinforced wallettes.

The RWX wallettes (scheme X) showed similar tension splitting cracks through the thickness of the Wallette (Figure 6c). As the tests continued, layer crushing of the bricks and some cracks in the mortar joints were observed under the loading shoes at the face shell of the specimen. Similar findings were also reported by Luccioni et al. [6]. The proposed diagonal reinforcing of the wallette effectively controlled diagonal cracks, leading to a more uniform distribution of cracks in terms of size and amount in the RWX and RWH specimens compared to the RWI wallettes. The number of cracks appearing on the surface of the reinforced RWI wallettes was significantly more than those of the other reinforced wallettes. The reinforcement of the compressed diagonal only improved the crack size, without any enhancement to the wallette’s strength. The numerical model also predicted this behavior: the extent of cracking predicted numerically for RWI was higher than that for the RWX wallette, which agrees with the experimental results.

Debonding at the CFRP–masonry interface was primarily observed in the regions proximal to the loaded corners. Debonding occurred for the portion of CFRP fabric located perpendicular to the compressed diagonal due to the high tensile stress at this location. The same mode in masonry walls reinforced with FRP strips has been observed by [50]. The RWX specimen remained almost totally uncracked, although the numerical crack pattern was essentially the same as that obtained from the test. Debonding failure at FRP-to-masonry interfaces occurred mostly through the cracking of the layer of the masonry, consistent with observations of masonry walls reinforced with FRP strips. The numerical debonding failure mechanism closely resembles that obtained from the experimental tests.

4.2. Convergence Study

4.2.1. Influence of Mesh Density

To determine an appropriate mesh density that balanced accuracy and computational cost, a mesh sensitivity study was performed. Recognizing the anticipated high stress concentrations in the mortar joints, the meshing of the joints was made more densely than the brick units. Three distinct mesh densities were evaluated: 25/8 mm (brick/mortar), 20/5 mm, and 10/2.5 mm. As shown in

Table 5. Mesh size and numerical result comparison.

Mesh Size (Brick/Mortar) (mm)	Numerical Shear Stress (MPa)	Experimental Shear Stress (MPa)	Difference (%)	Computation Time (Seconds)
25/8	0.6	0.56	5.3	5400
20/5	0.57	0.56	1.75	7200
10/2.5	0.56	0.56	0	14,000

, the numerical shear stress converged as the mesh size decreased, with the finest mesh (10/2.5 mm) predicting a shear stress of 0.56 MPa, matching with an experimental shear strength of 0.56 MPa with a difference of 0% and a 14,000 s computation time. This compares favorably to results with 0.57 MPa stress; the difference is around 1.75 while it has an almost 2,5 decreased computation time compared to the 10/2.5 mm run. The computation time generally increases with denser meshes; however, it should be mentioned that the intermediate mesh had a high degree of computational efficiency. Given the minimal gain in accuracy from 20/5 MPa to 10/2.5 MPa, and in consideration of memory costs and computational capacity, the mesh size of 20/5 MPa was selected for this study to optimize all aspects of the model.

4.2.2. Influence of Fracture Energy

The sensitivity analysis (

Table 6. Sensitivity of FE model to Mode-I fracture energy and dilation angle.

Parameter	Value (Units)	Ultimate Load (kN)	Failure Mode	Crack Pattern (Description)
Mode-I Fracture Energy	0.005 N/mm	56.3	Tension splitting failure Debonding
	0.018 N/mm (Baseline)	57.58	Tension splitting failure Debonding	Similar to baseline
	0.020 N/mm	57.8	Tension splitting failure Debonding	Baseline crack pattern
				Similar to baseline
Dilation Angle	0 degrees	57.5	Tension splitting failure Debonding	Baseline crack pattern
	5 degrees	58.2	Tension splitting failure Debonding	Similar crack pattern to the baseline with a minor modified crack location

) indicated that the model exhibited relatively low sensitivity to variations in Mode-I fracture energy at the brick–mortar interface across the range of 0.005 to 0.020 N/mm. The ultimate load changed little, with most changes ranging from one to two percent, and the failure mode (tension splitting failure and debonding) and crack pattern remained qualitatively similar. However, the dilation angle displayed a slightly more apparent, but still ultimately small, impact. Increasing the dilation angle by 5 degrees resulted in a minor increase in the ultimate load (approximately 1.5%) and wider crack propagation, while decreasing the dilation angle resulted in the failure of the structure.

4.2.3. Accounting for Size and Confinement Effects

While our model utilizes a 400 × 400 mm² panel, a scale smaller than typical real-world applications, this choice was influenced by a need for computational efficiency and experimental validation and still is within the reasonable limits of the recommendations. The American standard ASTM E519 [16] and the RILEM recommendation LUMB6 [28] differ in their prescriptions for panel size and depth of the loading shoe.

To assess the potential impact of our smaller model size, we considered the findings of [51,52], who conducted a numerical investigation of panel size and loading shoe depth on wallette shear strength predictions. Their study, comparing panel widths ranging from 0.4 m to 1.4 m, revealed a negligible size effect for panels 0.4 m wide or greater, with a constant ${\alpha }_{calc}$ (the coefficient α corresponding to each stress component at the center of the panel), applying to panels from 0.6 m to 1.4 m. Additionally, Segura et al. [52] found that, for loading shoe depth ratios ranging from 1/12th to 1/7th of the panel side length, the ${\alpha }_{calc}$ coefficient remained relatively constant at around 0.4, suggesting minimal influence within this range. The selected model size falls within a range that exhibits negligible size effects, and measurements and analysis must be applied across other materials to further enhance experimental credibility and reproducibility. Therefore, future models should also incorporate a sensitivity study.

4.2.4. Stress–Strain Curves

The shear stress–strain curves obtained experimentally for all the wallettes are given in Figure 7. It can be observed that the use of CFRP strips improves the post-peak behavior and delays the failure of the reinforced specimens in comparison with the unreinforced specimens. Here, the stress–strain curve for the unreinforced masonry wallette indicates that the wallette exhibited an approximately linear shear stress–strain response until cracking followed by a slight increase in shear strength and deformation capacity before the rupture. Moreover, the control wallette failed at a lower load compared to all reinforced wallettes. The shear stress–strain curves derived from the numerical models were also compared with the experimental curves for each specimen. Figure 8 shows the stress–strain curves of the unreinforced and reinforced masonry wallettes, respectively. The deformation at the ultimate load of the reinforced wallettes was much higher than that of the unreinforced wallette. The bond–mortar interface has very limited deformation capacity, indicating a high degree of brittleness in this mode of failure. It should be noted that, for the reinforced wallette, the tensile stress is transferred from the masonry wallette to the CFRP fabric, and this leads to a reduction in masonry stress, thereby creating a ductile mode of deformation and a large energy absorption capacity. It is clear that the proposed model was able to predict the strength and stiffness of the unreinforced and reinforced masonry walls as well. The stiffness and strength from the analysis of the RWX wallette are slightly less than the experimental results, while both the stiffness and strength of the RWI and RWH wallettes are slightly higher than the actual test values. It should be noted that the numerical model could predict the post-peak response of the wallettes and was in good agreement with the measured responses.

The RWI wallette showed highly brittle behavior. Therefore, as can be seen in Figure 8, the post-peak behavior of the RWI wallette is not exactly the same as the experimental results. In this reinforced wallette configuration, the numerical model shows a more brittle behavior compared with experimental, and the load dropped significantly in this case. As can be seen from Figure 8, the RWX wallette has the highest stiffness compared with two other wallettes. However, the RWH wallette showed the same stiffness as the CW. Therefore, the reinforcement of the wallette with scheme H (RWH) had no effect on the stiffness of the wallette but showed a significant increase in strength. Thus, the increase in strength is highly noticeable, while stiffness is kept practically unchanged. The deformability of both the RWH and RWI wallettes increased compared to the RWX wallette. As can be seen in Figure 8, the RWH wallette has the highest shear strength compared with the two other wallettes. Subsequently, the CFRP reinforcement in this wallette prevented the sudden drop in load as cracks developed. This is evident from the smoother shear stress curve. No improvement in strength was achieved with CFRP reinforcement in the wallettes reinforced with scheme H.

The excellent agreement between the numerical and experimental results provides confidence in the significant effect of the proposed reinforcement details. The improved performance of the wallettes with different reinforcement configurations, compared with the unreinforced wallette, was demonstrated through the numerical study, showing good agreement with the experimental study. The configuration of the reinforcement, the reinforcement ratio of the fabric, and the failure modes that occurred in each wallette are summarized in

Table 7. Experimental evaluation of ultimate shear strength and ductility characteristics of unreinforced and CFRP-reinforced brick masonry wallettes.

Panels	Fibers (%)	Fmax (kN)	Ultimate Stress τ (MPa)	E (MPa)	G (MPa)	Axial Secant Stiffness at Failure (EA/L) kN/mm	μ	μu/μo	Failure Mode
CW	-	35.06	0.56	195	78	20.5	1.16	-	-Diagonal shear cracking
RWH	54.68	57.58	1.85	200	80	21	2.02	144	-Tension splitting failure -Debonding of CFRP composite
RWX	54.06	55.27	1.82	320	128	33.6	2.1	153	-Tension splitting failure -Debonding of CFRP composite
RWI	22.91	45.60	1.58	257.5	103	27	1.9	138	-Tension splitting failure -Diagonal shear cracking -Debonding of CFRP composite

All panels were 400 × 400 × 105 (lbh) mm³; μ: ductility factor; (μ_u/μ_o): comparative performance ratio: unreinforced vs. CFRP-reinforced masonry walls; fibers (%): proportion of wall surface area covered by carbon fiber reinforcement.

. These results indicate that the control wallettes had a failure load of 35 kN, while the failure loads for the strengthened scheme I, scheme H, and scheme X wallettes were 51 kN, 58 kN, and 55 kN, respectively. The CFRP strengthening increased the shear stress from 185% to 230% compared to the unreinforced wallettes. Moreover, the improvement in ductility for the strengthened wallettes ranged from 138% to 153%.

The RWX and RWH walls exhibited better shear behavior, with increases of up to 230%; these reinforcement configuration schemes also increased ductility, μ, by up to 150%, which proves that how the reinforcing composites are arranged has a very important effect on the local behavior of the structure because the arrangement needs to distribute stress and inhibit degradation. The average axial stiffness (AE/l) of the retrofitted specimens was 33.6 kN/mm, compared to 20.5 kN/mm for the control specimens. Retrofitting increased the axial stiffness of the wallettes by up to 160%. As can be seen in

Table 8. Validation of numerical models against experimental data for unreinforced and CFRP-reinforced masonry wallettes.

Panels	Fibers (%)	Numerical Shear Stress (MPa)	Experimental Shear Stress (MPa)	Error (%)
CW	0	0.57	0.56	1.75
RWH	54.68	1.613	1.85	4.10
RWX	54.06	2.117	1.82	0.04
RWI	22.91	1.544	1.58	2.30

, the numerical results are generally in good agreement with the test results for the different configurations.

4.2.5. Quantitative Analysis of Reinforcement Effect

The results presented in

Table 9. Influence of stiffnesses to energy dissipation and max shear stress from all configurations.

Curve	Initial Stiffness (MPa)	Max. Shear Stress (MPa)	Shear Strain at 75% of Max. Shear Stress	Stiffness Reduction Rate	Energy Dissipation at Failure (J)
CW	250	0.56	0.0016	0.0025	58.8
RWX	540	1.68	0.0015	1.5	95
RWI	520	1.45	0.0025	0.7	65
RWH	900	1.77	0.0032	1.8	486

reveal notable differences in the mechanical behavior of the unreinforced control wallette (CW) compared to the CFRP-reinforced configurations (RWX, RWI, and RWH). The most striking observation is in the initial stiffnesses. The RWH, RWX, and RWI wallettes each saw increases, respectively, compared to the CW. Each of these walletes had the benefit of an improved load capacity, with the maximum shear test results for the RWH wallette and the minimum for the CW.

The higher energy dissipation capacity of the RWH and RWX schemes suggests that these configurations are more effective at absorbing energy during seismic events. This is important for preventing collapse and improving the overall safety of the structure.

The Fully Ruptured Ratio (FRR) in this study is defined as the ratio of the total volume of fully damaged regions to the overall volume of the wallette. This volume-based metric is particularly appropriate for our 3D FE model as it captures the full three-dimensional nature of the damage within the masonry wall. While surface-based metrics might be more intuitive for visualizing crack patterns, they may not fully account for the extent of damage that extends into the thickness of the wall. Using a volume-based FRR enables us to obtain a more comprehensive assessment of the walls’ overall damage states and the effectiveness of the CFRP reinforcements in mitigating failure.

The results presented in

Table 10. Quantification of damage: Volume of fully damaged regions and Fully Ruptured Ratio.

Configuration	Component	Number of Fully Damaged Elements	Volume of Fully Damaged Region (cm3)	Overall Volume (cm3)	$\mathbf{FRR}=\frac{{\mathit{V}}_{\mathit{f}\mathit{d}}}{{\mathit{V}}_{\mathit{T}\mathit{o}\mathit{t}}}$ ×100 (%)
CW (Control)	Brick	107	856	17,400,346	0.023
	Mortar	25,725	3215
RWI (Scheme I)	Brick	61	488	17,400,346	0.00280
RWH (Scheme H)	Brick	57	476	17,400,346	0.0027
RWX (Scheme X)	Brick	23	184	17,400,346	0.00105

show the Fully Ruptured Ratios (FRRs), representing the percentages of the overall volumes that are fully damaged, and provide insights into the effectiveness of each CFRP reinforcement scheme. The control wallette (CW) exhibited a significantly higher FRR (0.023%) compared to the reinforced wallettes, indicating a greater proportion of fully damaged material at failure. The calculation for the control wallette also included its total volume of brick units, which is why the control volume is also higher for the brick units. Furthermore, note that the RWI, RWH, and RWX results do not account for the damages in mortar. This shows the high focus on brick as opposed to mortar in this analysis.

While all the CFRP reinforcement configurations reduced the FRR compared to the control, the RWX wallette exhibits the lowest FRR because it has a configuration in which there is a reduction in tension splitting cracks and a more uniform size of cracks.

5. Shear Design of FRP-Reinforced URM Walls

The shear resistance of a URM wall strengthened with FRP is the sum of the URM’s inherent shear resistance and the additional shear resistance provided by the FRP reinforcement. The FRP’s contribution to the shear strength of a polymer–masonry wall composite is calculated using Equation (6), which considers the FRP’s contribution to the lateral resistance $F_{FRP}$ of the URM specimen including:

The reinforcement ratio, ${\rho }_h$, of the FRP in the horizontal direction;

The modulus of elasticity, $E_{FRP}$, of FRP;

The ultimate strain, ${\epsilon }_{tu}$, of FRP;

An efficiency factor, $k$;

The wall thickness, $t$;

The wall length, $L$.

$$F_{FRP}={\rho }_h{E}_{FRP}{\epsilon }_{tu}k t L$$

(6)

Various values have been proposed for the efficiency factor. Triantafillou, in his work on reinforced concrete beams, derived an empirical polynomial function relating the strain in the FRP at shear failure to the axial rigidity of the composites. This polynomial has been applied to masonry walls. A study involving diagonal tension tests on two GFRP-retrofitted specimens [53] suggests a $k$ value of 0.3. Zhao et al. [54], through static cyclic tests, proposed a value of 0.2 for precracked specimens and 0.3 for uncracked specimens.

Several models have been used to calculate the shear strength of FRP-reinforced URM walls. These models all assume that the total shear strength of the reinforced wall is the sum of the URM wall’s shear strength and the FRP’s contribution. The Triantafillou model [55] is designed for walls reinforced with narrow-strap FRP laminates. The model calculates the FRP contribution to shear strength based on the effective FRP strain, which is a function of the FRP area fraction in the horizontal direction, the modulus of elasticity of the FRP, and the partial safety factor for the FRP in uniaxial tension. The Garbin et al. model [56] adopts distinct equations for FRP strips and FRP laminates, with the FRP contribution to shear strength calculated based on effective design strength, the cross-sectional area of FRP, and both reinforcement spacing and the effective stress in the FRP. CNR-DT 200/2013 [57], a standard specifically tailored for masonry walls reinforced with FRPs, recognizes that the primary failure mode is the loss of adhesion between the FRP and the masonry. As such, its allowable stress is directly influenced by the length of the bonded area. This standard defines the maximum length over which effective anchoring can be achieved. The AC 125 Model [58] is specifically designed for rectangular-cross-section walls reinforced with FRP attached on one or both sides. The model calculates the FRP contribution to shear strength using the thickness of the FRPs, the height of the wall, the angle of fiber orientation, and the tensile strength of the composite material. Finally, the ACI 440.7R-10 [59] model considers shear-controlled fracture and defines the FRP reinforcement contribution through a reduction coefficient of the bond. The total force that the FRP system can exert on the wall is determined based on the height and length of the panel, the center-to-mid-interval FRP reinforcement, the cross-sectional area of the FRP reinforcement, and the effective stress in the FRP. The effective stress is calculated as a function of the effective strain, $k$ function, and the ultimate fracture strain of the FRP.

This study investigates the shear strength of unreinforced masonry (URM) walls subjected to diagonal compression. The results from this analysis are compared against existing design equations (

Table 11. Experimental shear strength of URM walls compared to design predictions.

Sample Code	Experimental Results (τ) (MPa)	ACI 440.7R-10 Model (MPa)	AC 125 Model (MPa)	Garbin et al.’s Model [56] (MPa)	Triantafillou Model (MPa)	CNR-DT 200/2013 Model (MPa)
URM (CW)	1.080	0.95	0.95	0.95	0.95	0.95
RWH	1.613	1.65	1.67	1.78	1.53	1.65
RWX	2.117	2.03	2.25	2.65	2.03	2.03
RWI	1.544	1.61	1.68	1.7	1.68	1.6
Average absolute error (%)		4.27	8.8	25.5	10.1	3.62

), specifically focusing on the shear strength of the reinforced masonry. The following regulations and design models were used for comparison: the Triantafillou model [55], Garbin et al.’s Model [56], CNR-DT 200/2013 [57], AC 125 Model [58], and the ACI 440.7R-10 [59]. The results were statistically processed to determine the average absolute error percentage.

Table 11 presents a comparison of the different models for predicting the shear strength of FRP-reinforced masonry walls, based on the experimental results. The ACI 440.7R-10 model exhibits the lowest average absolute error (4.27%), suggesting that it provides the most accurate predictions among the models examined. The AC 125 Model, with an average error of 8.8%, shows less accuracy. In contrast, Garbin et al.’s model [56] demonstrates substantial discrepancies, with an average error of 25.5%. The Triantafillou model exhibits a moderate average error of 10.1%, while the CNR-DT 200/2013 model shows the second-lowest average error (3.62%), indicating its strong performance in predicting tensile strength. These findings suggest that the ACI 440.7R-10 model and CNR-DT 200/2013 model are more reliable for predicting the tensile strength of FRP-reinforced masonry walls. However, the significant deviations observed with Garbin et al.’s model underscore the importance of selecting the appropriate model for specific applications and understanding its limitations.

6. Conclusions

This study examined polymer–masonry composites by investigating the in-plane behavior of unreinforced masonry wallettes strengthened with three different configurations of carbon fiber-reinforced polymer. A micro-modeling approach was employed, incorporating damage plasticity and surface-based cohesive-contact-interface methods. The findings revealed several key insights:

• CFRP strip reinforcement significantly enhanced both the shear bearing capacity and ductility of the masonry wallettes.
• Reinforcement with diagonal CFRP strips bonded orthogonally eliminated the shear-dominated failure mode.
• While reinforcing only the compressed diagonal of the wallettes (RWI) resulted in a 138% increase in ductility, there was no significant improvement in shear capacity. This indicates that simply reinforcing the compressed diagonal alone is insufficient for enhancing strength, and a more comprehensive approach is needed.
• Understanding the debonding mechanism is crucial for predicting the failure behavior of reinforced walls. Debonding typically occurs when cracks develop within the masonry surface, leading to failure the of the bond between the FRP and the masonry.
• The strength and ductility of masonry walls are significantly affected by the placement of the CFRP fabric. Further investigation using different reinforcement configurations is recommended to determine the most effective and economical approach to retrofitting masonry wallettes.
• The unreinforced masonry models accurately represent the brittle behavior of URM walls. Furthermore, the validated models exhibit enhanced ductility and resistance observed in CFRP-reinforced walls. These numerical simulations can effectively assess and predict reinforcement effects.

The CDP model and cohesive approach employed in this numerical study were proven effective in modeling the behavior and failure mechanisms of reinforced masonry walls. The developed numerical models show promising practical applications in predicting the progression of damage in CFRP-reinforced masonry walls subjected to in-plane loading. Furthermore, the finite element model demonstrated strong agreement with experimentally observed crack patterns, accurately predicting crack formation within the mortar layer, bricks, and at the CFRP-to-masonry interface. This validation supports the accuracy of the FE model. However, while the FE model accurately predicted some microcracks in hard-to-see regions, it did not capture all the observed fractures at the brick–mortar joints. This highlights the need for further refinements in order to model microcracking behavior with greater precision.

While this study provides valuable insights into the behavior of CFRP-reinforced masonry walls, it is important to acknowledge its limitations. The study only investigated three specific reinforcement configurations (X, I, and H), and the results may not be generalizable to other configurations. The study also used specific types of bricks, mortar, and CFRPs, and the results may not apply to other materials. The parameters were calibrated to match the experimental results, which may have led to bias in the study. Finally, the study only tested walls under diagonal compression loading, and the results may not apply to other types of loading. Future research should address these limitations by conducting more parametric studies, using a wider range of materials, and developing more sophisticated FE models.

Although the developed models demonstrate efficiency and accuracy within the studied cases, further work is needed. Future research should prioritize a thorough parametric study on the factors influencing debonding, such as adhesive properties, interface roughness, and bonding techniques, to better capture fracture behavior and understand the debonding failure of the FRP-to-masonry interface under various stress conditions.

The accurate prediction of the behavior of CFRP-strengthened masonry requires the careful modeling of the FRP–masonry interface, particularly when considering debonding, a significant failure mode. While cohesive interface elements with well-defined parameters are recommended for modeling debonding, engineers should also consider design strategies to mitigate debonding, such as increased bond areas or adhesives with higher fracture energies. Furthermore, given the limitations of the existing design equations, FEA simulations can serve as a valuable tool, especially for complex geometries and loading conditions.